Question: Determine how many solutions exist for the system of equations. ${-3x+y = -9}$ ${3x-y = 1}$
Solution: Convert both equations to slope-intercept form: ${-3x+y = -9}$ $-3x{+3x} + y = -9{+3x}$ $y = -9+3x$ ${y = 3x-9}$ ${3x-y = 1}$ $3x{-3x} - y = 1{-3x}$ $-y = 1-3x$ $y = -1+3x$ ${y = 3x-1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x-9}$ ${y = 3x-1}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.